Geometry of measures and applications

University Of Washington, Seattle WA

Investigators

Abstract

When dipping a wire frame in a solution of soap suds one produces a thin soap film. Mathematically this is a very interesting object. It is closely related to the solution of the Plateau problem, which requires one to find a surface of minimal area that spans a given contour in space. (This classical problem is an example of those that one encounters in the calculus of variations.) The area of a surface can be understood as a measurement of energy. The basic guiding principle is that minimizing its energy will lead to a stable configuration in any physical system. In this project the principal investigator addresses questions concerning the minimization of energies, questions that take into account noise and small fluctuations of the phenomena being modelled. The hope is that this theory will be better suited than existing ones to address minimization questions that arise in nature. The principal investigator's goal in the project is to show that "almost minimizers," which are minimizers to noisy variational problems, inherit some of the properties of minimizers of the same functional minus the noise. This research requires the use of tools from the calculus of variations, harmonic analysis, partial differential equations, potential theory, and geometric measure theory. The project will build bridges between the aforementioned areas, hopefully transforming them in the process through the influx of new ideas. The expectation is that these new ideas will, in particular, find applications in other variational problems with free boundaries.